Processing for compression - The MPEG - page 86

Figure 3.48: This wavelet decoder can perfectly recover the transformed data of Figure 3.47 .
Provided that the decoding filters have responses which are appropriate to the responses of the encoding filters,
the output sample stream will be identical to the original. This is known as perfect reconstruction and the practical
use of wavelets relies upon finding sets of four filters which combine to display this characteristic.
Figure 3.47 (b) shows that stages are cascaded in such a way that the low-pass process iterates. At each stage the
number of samples representing the original block has halved, but exactly the same filter pair is employed. As a
result the wavelet at each stage appears to have become twice as long with respect to the input block.
As a result of this cascading, wavelets are naturally and fundamentally scaleable. At the end of a cascade is a
heavily band-limited signal. Adding coefficients from the next frequency band doubles the resolution available.
Adding coefficients from the next band further doubles it.
The wavelet transform outputs the same amount of data as is input. Figure 3.49 (a) shows a one-dimensional
example of a four-stage cascade into which sixteen samples are fed. At the first stage eight difference values are
created. At the next, four difference values are created. At the next there are two. Finally a single difference value
is created along with a single output value from the sequence of four low-pass filters which represents the average
brightness of the 16 pixels. The number of output values is then:
8 + 4 + 2 + 1 differences + 1 average = 16
Figure 3.49: (a) A four-stage one-dimensional wavelet cascade. (b) When used with two-dimensional pixel arrays,
each wavelet stage divides the data into one quarter and three quarters. (c) Cascading the process of (b).
Like other transforms, the wavelet itself does not result in any compression. However, what it does do is to
represent the original information in such a way that redundancy is easy to identify.
Figure 3.50 shows that that a set of wavelets or basis functions can be obtained simply by dilating (scaling) a single
wavelet on the time or space axis by a factor of two each time. Each wavelet contains the same number of cycles
such that as the frequency reduces, the wavelet gets longer. In a block of fixed size, a large number of short
wavelets are required to describe the highest frequency, whereas the number of wavelets at the next resolution
down would be halved. The desirable result is that as frequency rises, the spatial or temporal resolution increases.
Figure 3.51 (a) shows the fixed time/frequency characteristic of the Fourier transform. This can be improved by
using window sizes which are a function of frequency as in(b). The wavelet transform in (c) has better temporal
resolution as frequency increases due to the dilation of the basis function.